Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

Abstract

In the current investigation, both unforced and forced Duffing–Van der Pol oscillator (DVdPV) oscillators with a strong nonlinearity and external periodic excitations are analyzed and investigated analytically and numerically using some new and improved approaches. The new approach is constructed based on Krylov–Bogoliubov–Metroolsky method (KBMM). One of the most important features of this approach is that we do not need to solve a system of differential equations, but only solve a system of algebraic equations. Moreover, the ease and faster of applying this method gives high-accurate results and this approach is better than many approaches in the literature. This approach is applied for analyzing (un)forced DVdP oscillators. Also, some improvements are made to He’s frequency-amplitude formulation in order to solve unforced DVdP oscillator to obtain high-accurate results. Furthermore, the He’s homotopy perturbation method (He’s HPM) is employed for analyzing unforced DVdP oscillator. The comparison between all mentioned approaches is carried out. The application of our approach is not limited to (un)forced DVdPV oscillators only but can be applied to analyze many higher-order nonlinearity oscillators for any odd power and it gives more accurate results than other approaches. Both used methods and obtained approximations will help many researchers in general and plasma physicists in particular in the interpretation of their results.